Algebra Examples

$x = 2 + \sqrt{x^{2} - 1}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 + \sqrt{x^{2} - 1} = x$

Step 2

Subtract $2$ from both sides of the equation.

$\sqrt{x^{2} - 1} = x - 2$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} - 1}}^{2} = {(x - 2)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 1}$ as ${(x^{2} - 1)}^{\frac{1}{2}}$ .

${({(x^{2} - 1)}^{\frac{1}{2}})}^{2} = {(x - 2)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(x^{2} - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(x^{2} - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} - 1)}^{\frac{1}{2} \cdot 2} = {(x - 2)}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(x^{2} - 1)}^{\frac{1}{2} \cdot 2} = {(x - 2)}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(x^{2} - 1)}^{1} = {(x - 2)}^{2}$

Step 4.2.1.2

Simplify.

$x^{2} - 1 = {(x - 2)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(x - 2)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$x^{2} - 1 = (x - 2) (x - 2)$

Step 4.3.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$x^{2} - 1 = x (x - 2) - 2 (x - 2)$

Step 4.3.1.2.2

Apply the distributive property.

$x^{2} - 1 = x \cdot x + x \cdot - 2 - 2 (x - 2)$

Step 4.3.1.2.3

Apply the distributive property.

$x^{2} - 1 = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} - 1 = x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 4.3.1.3.1.2

Move $- 2$ to the left of $x$ .

$x^{2} - 1 = x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2$

Step 4.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$x^{2} - 1 = x^{2} - 2 x - 2 x + 4$

Step 4.3.1.3.2

Subtract $2 x$ from $- 2 x$ .

$x^{2} - 1 = x^{2} - 4 x + 4$

Step 5

Solve for $x$ .

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Step 5.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 4 x + 4 = x^{2} - 1$

Step 5.2

Move all terms containing $x$ to the left side of the equation.

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Step 5.2.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 4 x + 4 - x^{2} = - 1$

Step 5.2.2

Combine the opposite terms in $x^{2} - 4 x + 4 - x^{2}$ .

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Step 5.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$- 4 x + 4 + 0 = - 1$

Step 5.2.2.2

Add $- 4 x + 4$ and $0$ .

$- 4 x + 4 = - 1$

Step 5.3

Move all terms not containing $x$ to the right side of the equation.

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Step 5.3.1

Subtract $4$ from both sides of the equation.

$- 4 x = - 1 - 4$

Step 5.3.2

Subtract $4$ from $- 1$ .

$- 4 x = - 5$

Step 5.4

Divide each term in $- 4 x = - 5$ by $- 4$ and simplify.

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Step 5.4.1

Divide each term in $- 4 x = - 5$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 5}{- 4}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $- 4$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 5}{- 4}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 4}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Dividing two negative values results in a positive value.

$x = \frac{5}{4}$

Step 6

Exclude the solutions that do not make $x = 2 + \sqrt{x^{2} - 1}$ true.

$No solution$